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Chapter 6: Problem 265

In the following exercises, solve each proportion. $$ \frac{n}{91}=\frac{8}{13} $$

Short Answer

Expert verified
n = 56

Step by step solution

01

Understand the Proportion

The given proportion is \ \( \frac{n}{91}=\frac{8}{13} \). This means that the two ratios are equal, and we need to find the value of \( n \).
02

Cross Multiply

To solve the proportion, cross-multiply the terms. This means multiply the numerator of one ratio by the denominator of the other ratio and set the products equal to each other. \ \( n \times 13 = 8 \times 91 \)
03

Simplify the Equation

Perform the multiplication on the right side of the equation: \ \( 13n = 728 \) (since \( 8 \times 91 = 728 \)).
04

Solve for \( n \)

To isolate \( n \), divide both sides of the equation by 13: \ \( n = \frac{728}{13} \)
05

Simplify the Result

Simplify the fraction to find the value of \( n \): \ \( n = 56 \) (since \( 728 \div 13 = 56 \)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Ratios
A ratio compares two quantities to show their relationship. In the proportion \(\frac{n}{91} = \frac{8}{13}\), the ratios compare values that are treated as equal. Ratios can be written in different forms:
  • Fraction form: \( \frac{a}{b} \)
  • Using a colon: a:b
  • Using words: 'a to b'
A proportion means two ratios are equal. To solve problems involving proportions, you need to work with these ratios correctly.
Cross-Multiplication
Cross-multiplication is a method used to solve proportions. By cross-multiplying, you eliminate the fractions which makes it easier to solve for the unknown variable. In the given proportion \(\frac{n}{91} = \frac{8}{13}\), cross-multiplying means:
  • Multiplying the numerator of the first ratio by the denominator of the second ratio: \(n \times 13\)
  • Multiplying the denominator of the first ratio by the numerator of the second ratio: \(91 \times 8\)
Set the two products equal: \(13n = 728\). This simplifies the equation and helps us find the value of \'n\.
Simplifying Equations
Once you have an equation like \(13n = 728\), the next step is simplifying it to solve for the variable. Simplifying means performing mathematical operations to make the equation easier to handle. Here are basic steps:
  • Perform the multiplication or division on both sides of the equation if required.
  • Combine like terms if there are any.
So from \(13n = 728\), we can see that performing multiplication is already done, making our next task to solve for \'n\.
Isolation of Variables
To solve for a variable, you need to isolate it on one side of the equation. This often involves:
  • Adding or subtracting terms from both sides.
  • Multiplying or dividing both sides by the same number.
For our equation \(13n = 728\), we isolate \ by dividing both sides by 13:
  • \(n = \frac{728}{13}\)
After simplifying, \ equals 56. Isolating variables is a key step in solving equations, ensuring you get the accurate value of the unknown.

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